1. Objective Probabilities
The Sociology of Risk
Objective Probabilities
Goals:
To understand how to calculate objective probabilities
To recognize that our understanding of objective probabilities is shaped by our perceptions

2. Risk and Uncertainty Arnoldi defines ‘risk’ as potential dangers
Risk usually refers to negative events, but ‘risk’ is sometimes used in a generic ways for positive and negative events
Probability is implied in the definition (i.e., ‘potential’)
‘Risk’ refers to a situation where the probability of the event can be calculated
‘Uncertainty’ refers to a situation where the probability of the event cannot be calculated
There is a difference between objective and subjective probabilities

3. Subjective probability
Subjective probabilities depend on the person making the assessment
Compare a coin flip to a horse race – what is the difference?
Some events/games are not repeatable

4. Objective Probability
“An objective probability is a probability that everyone agrees on” (Amir D. Aczel Chance: A Guide to Gambling, Love, the Stock Market, & Just About Everything Else.)
Probability is the ratio of the number of times the desired outcome can occur relative to the total number of all outcomes that can occur over the long run.
Probabilities are often expressed as ratios and/or proportions.

5. Coin example
The probability of a ‘heads’ on one flip of an honest coin=1/2=0.5
The flip of a coin is a purely random event – we cannot predict the outcome of one flip with certainty
We can predict that the proportion of heads over many flips is 0.5
As the number of flips increases, the proportion will center on the value of 0.5

6. Coin example Elementary outcomes: heads, tails
This is an equal probability process

7. Dice example
The probability of a ‘6’ on one roll of an honest die=1/6=0.1667
The roll of a die is a purely random event – we can’t predict the outcome of any one roll
We can, however, predict that the proportion of 6s over many rolls is about
As the number of rolls increases, the proportion will center on

8. Dice example Elementary outcomes: 1, 2, 3, 4, 5, 6
This is an equal probability process

9. Basic rules for more complex events…
The ‘or’ rule (when to add)
The probability of a 1 or a 6 on one roll of an honest die=1/6+1/6=2/6=0.333
The ‘and’ rule (when to multiply)
The probability of a 6 and a 6 on the roll of two honest dice=1/6*1/6=1/36=0.0278

10. Two dice
36 Elementary outcomes
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
6,4
6,5
6,6
Sum
Die 1 1 2 3 4 5 6
Die 2 7 8 9 10 11 12
11 Possible totals
Expected frequency
Expected proportion 2 1
0.0278 3
0.0556 4
0.0833 5
0.1111 6
0.1389 7
0.1667 8 9 10 11 12
Sum 36

11. Coin example Four flips of an ‘honest’ coin and the number of heads
This is a binomial process
Both outcomes (heads and tails) are equally likely so there is a uniform distribution for one flip
There is NOT a uniform distribution for multiple flips – with 4 flips there are 5 possible outcomes but 16 ways to get them

12. How many heads on four flips of an honest coing?
Probability
0.0625
0.2500
0.3750
# Heads 1 2 3 4
Ways 6
TTTT
HTTT
HHTT
HHHT
HHHH
THTT
HTHT
HHTH
TTHT
HTTH
HTHH
TTTH
THHT
THHH
THTH
TTHH

13. Other Games – the Lottery
It is possible to calculate the objective probability of many games…
Lottery (6 balls numbered 1-54)
You need the 1 correct set of six numbers out of the 25,827,165 possible unique combinations of six numbers
…So don’t play the lottery

14. Roulette A roulette wheel has 38 buckets:
36 numbers (1-36)
0 (green)
00 (green)
2 colors for the 36 numbers: red and black
The house has a built-in advantage (because of how it sets the bets – for example, 35 to 1 for a single number); it wins over the long run

16. United States Roulette Rules
Bet
Pays
Probability Win
House Edge
Red 1
47.37%
5.26%
Black
Odd
Even
1 to 18
19 to 36
1 to 12 2
31.58%
13 to 24
25 to 36
Sixline (6 numbers) 5
15.79%
First five (5 numbers) 6
13.16%
7.89%
Corner (4 numbers) 8
10.53%
Street (3 numbers) 11
Split (2 numbers) 17
Any one number 35
2.63%

17. Blackjack
What makes Blackjack so interesting is that it is based on continuous probability; the probabilities actually change during play with each passing card…it is a game with a memory; this is what makes it possible to beat the casino (assuming an auto shuffler is not used after every hand with replacement)
Basic rules
Closest to 21 wins; tie=‘push’
Ace=1 or 11
2-9=face value
10, J, Q, K=10
Dealer must hit until sum totals 17, 18, 19, 20, or 21
Blackjack=21 on two cards; beats all but dealer blackjack, pays 3 to 2
Plays: hit, stay, split, double down (double bet for 1 card), surrender (you get 50% of your bet back), insurance (costs half of current wager, pays 2 to 1 if Blackjack 21)

18. Blackjack
The longer you play, the greater the likelihood that you will lose everything, but you can improve your chances by following basic strategy…

19. Blackjack Counting cards – the Hi-low System
Developed by MIT Professor Edward Thorp
Simulations show that when low cards (7 and under) are left in the deck, the odds favor the dealer; high cards (9 and up) favor the player
The Hi-lo system is based on a running tally, not memorizing every card
+1: 2-6
-1: 10, J, Q, K, Aces
7, 8, 9 are not counted
You increase your bet when the count is high
An equation determines how much to raise the bet (count/number of decks not seen); also incorporate house advantage

20. Probabilities are confusing
Calculating objective probabilities is not always easy, but it can be done
Despite this, many problems/games confound people because they are non-intuitive
Monty Hall Problem / Let’s Make a Deal
Contestant selects one of three doors
From the remaining two doors, the host selects one non-winning door
The contestant is asked: stay with their original selection or change doors?
What should they do?
Probability of picking the winning door is 1/3
Probability of not picking the winning door is 2/3
When one of the non-selected doors is revealed, the probability for the two non-selected is still 2/3
So the probability of switching and winning is 2/3
The probability of not switching and winning is still 1/3

21. Monte Hall
Door 1
Door 2
Door 3
result if switching
result if staying
Initial pick is always Door 1
Car
Goat
If stay, win 1/3
If switch, win 2/3
vos Savant, Marilyn (1990). "Ask Marilyn" column, Parade Magazine p. 16 (9 September 1990).

22. Paul the Octopus
Paul the octopus picked the winner of 8 straight soccer matches in 2010 FIFA World Cup
Is Paul some sort of soccer genius?

23. Paul the Octopus
# games correct in a row
Probability
How many animals would need to make predictions for 1 to be correct 8/8?
0.5 1 2 4 3 8 16 5 32 6 64 7
128
256
You didn’t hear about the other 255 animals that didn’t predict as well as Paul…

24. Perceptions of Objective Probabilities
Things casinos do:
Pictures of recent big winners with checks
Slots that flash, have sirens, and falling coins – these draw our attention
Use of chips instead of US Dollars
Biases:
We have selective memory
We suffer from hopeful thinking
We use faulty logic
Gambling versus counting
Good ‘cheats’ don’t always win; they utilize information to increase their chances
They must be subtle or it won’t work